# [EM] 3-candidate ratio space (Or how learnt to stop worrying and love linear algebra) (fwd)

David Catchpole s349436 at student.uq.edu.au
Tue Jun 1 22:06:46 PDT 1999

```Hey guys! Me stuffing my maths up yet again!
I was worried about not really understanding or trusting Saari's
derivation of the space of ratios of votes (especially as it fails
apparently to include the opinion of those who are indifferent between
options) so I tried to derive it myself, using a few
tricks from linear algebra... Problem is I get the following description
which I am having problems simplifying-

The space of the admissable ratios x (those who prefer candidate A to B
over those with some opinion of difference between A and B), y (blah blah
A to C) and z (blah blah B to C) is such that 0<=x,y,z<=1 and within this
restraint is the union of the following spaces-

Two families A: 2xyz-xy-yz+y-xz>=0 and B:2xyz-xy-yz+y-xz<=0

B(i) -yzx+yx-y+zx<=0 1+zx-z-x>=0
A(i) -yzx+zy-y+x>=0 -y-z+zy+1>=0
A(ii) -y+zy-yzx+yx+zx>=0
B(ii) -yzx+z+x-1<=0
B(iii) zx-yzx+zy-y<=0
A(iii) -yzx+yx-y+z>=0
A(iv) -y+x>=0 -2z-y+2zy+1>=0
B(iv) -y+x<=0 -2z-y+2zy+1<=0
A(v) x+z-1>=0 -2y+2zy-z+1<=0 -yx-zy+y+x+z-1>=0
B(v) x+z-1<=0 -2y+2zy-z+1>=0 -yx-zy+y+x+z-1<=0
A(vi) -2x+1>=0 z-y>=0
B(vi) -2x+1<=0 z-y<=0

```